Question

What is the antiderivative of ${e^{ - 3x}}$ ?

Answer 1

Hint : The function is given in exponential form. We will use a substitution method. The power of the exponent will be substituted and then we will proceed to find the integration or the antiderivative as mentioned.

Complete step by step solution:
Given that,
${e^{ - 3x}}$
To find the antiderivative means,
$\int {{e^{ - 3x}}} dx$
Now we will use substitution as ,
$- 3x = u$
Taking the derivative on both sides,
$- 3dx = du$
The value of dx is,
$dx = - \dfrac{{du}}{3}$
Now substitute in the original equation,
$= - \int {{e^u}\dfrac{{du}}{3}}$
Taking the constant ratio outside,
$= - \dfrac{1}{3}\int {{e^u}du}$
We know that the integration of the exponential function of this type is the function itself,
$= - \dfrac{1}{3}{e^u} + C$
Now replace the value of u,
$= - \dfrac{1}{3}{e^{ - 3x}} + C$
This is the correct answer.
So, the correct answer is “$- \dfrac{1}{3}{e^{ - 3x}} + C$”.

Note : Note that the antiderivative is nothing but the integral. When exponential function is concerned, we know that if it is of the form $\int {{e^x}dx}$ then the answer is definitely the function only. But if it is like if the exponent is other than this its better to use a method of substitution.
Also don’t forget to write the minus sign here in this case.
Also write the constant C at the end.